If the common tangents to the parabola, x 2 = 4 y and the circle, x 2 + y 2 = 4 intersect at the point P , then the distance of P from the origin (units), is:

Step-by-step solution

Let y = m x + c be the common tangent. Then c 2 = 4 1 + m 2   . . . ( 1 ) (condition of tangency for circle) Solving with x 2 = 4 y , we get x 2 = 4 m x + c i.e. x 2 - 4 m x - 4 c = 0   . . . ( 1 ) Being a tangent, ( 1 ) must have equal roots. ⇒ - 4 m 2 = 4 1 - 4 c ⇒ m 2 = - c   . . . ( 2 ) From 1   &   2 ,   c 2 = 4 - 4 c   &   c < 0 ⇒ c 2 + 4 c - 4 = 0 ⇒ c = - 2 2 - 2 (As c < 0 , so c ≠ 2 2 - 2 ). So, both tangents have common y